Subgroups — C₂³×C₂₃×#97 #19207677

Order 5888.

4126 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₂₃	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#97	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₃	C₄	1	2	yes	—
5	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₄	1	32	yes	complemented
7	C₂³×C₈×C₂₃	C₂²	1	28	yes	—
8	C₂²×C₄²×C₂₃	C₄	1	14	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	14	yes	—
10	C₂×C₂₃×#97	C₂²	1	112	yes	direct product factor
11	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	—	2	16	no	—
13	C₂³×C₄×C₂₃	C₂×C₄	1	28	yes	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	7	yes	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	16	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	7	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	224	yes	complemented
18	C₂²×C₈×C₂₃	C₂³	1	56	yes	—
19	C₂×C₄²×C₂₃	C₂×C₄	1	84	yes	—
20	C₂×C₄²×C₂₃	C₂³	1	28	yes	—
21	C₂₃×#97	C₂³	1	64	yes	direct product factor
22	C₂⁴×C₂₃	C₄²	1	8	yes	—
23	C₂⁴×C₂₃	M₁₆	1	16	yes	—
24	C₂⁴×C₂₃	C₂²×C₄	1	7	yes	—
25	C₂²×C₄×C₂₃	—	2	112	no	—
26	C₂²×C₄×C₂₃	C₄²	1	56	yes	—
27	C₂²×C₄×C₂₃	C₂²×C₄	1	56	yes	—
28	C₂²×C₄×C₂₃	C₂²×C₄	1	21	yes	—
29	C₂²×C₄×C₂₃	C₂²×C₄	1	56	yes	—
30	C₂²×C₄×C₂₃	C₂⁴	1	7	yes	—
31	C₂×C₈×C₂₃	C₂²×C₄	1	224	yes	complemented
32	C₂×C₈×C₂₃	C₂⁴	1	16	yes	—
33	C₄²×C₂₃	C₂²×C₄	1	56	yes	—
34	C₄²×C₂₃	C₂⁴	1	8	yes	—
35	C₂³×#97	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₂₃	#97	1	64	yes	direct product factor
37	C₂³×C₂₃	C₂×C₄²	1	28	yes	—
38	C₂³×C₂₃	C₂×M₁₆	1	56	yes	—
39	C₂³×C₂₃	C₂³×C₄	1	7	yes	—
40	C₂×C₄×C₂₃	—	2	112	no	—
41	C₂×C₄×C₂₃	C₂×C₄²	1	84	yes	—
42	C₂×C₄×C₂₃	C₂³×C₄	1	16	yes	—
43	C₂×C₄×C₂₃	C₂³×C₄	1	28	yes	—
44	C₂×C₄×C₂₃	C₂³×C₄	1	7	yes	—
45	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
46	C₈×C₂₃	C₂³×C₄	1	32	yes	complemented
47	C₂⁴×C₈	C₂×C₂₃	1	2	yes	—
48	C₂³×C₄²	C₂×C₂₃	1	1	yes	characteristic
49	C₂²×#97	C₂×C₂₃	1	28	yes	direct product factor
50	C₂²×C₂₃	C₂×#97	1	112	yes	direct product factor
51	C₂²×C₂₃	C₂²×C₄²	1	14	yes	—
52	C₂²×C₂₃	C₂²×M₁₆	1	28	yes	—
53	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
54	C₄×C₂₃	—	2	16	no	—
55	C₄×C₂₃	C₂²×C₄²	1	14	yes	—
56	C₄×C₂₃	C₂⁴×C₄	1	2	yes	—
57	C₂⁴×C₄	C₄×C₂₃	1	2	yes	—
58	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
59	C₂³×C₈	C₄×C₂₃	1	32	yes	complemented
60	C₂³×C₈	C₂²×C₂₃	1	28	yes	—
61	C₂²×C₄²	C₄×C₂₃	1	14	yes	—
62	C₂²×C₄²	C₂²×C₂₃	1	14	yes	—
63	C₂×#97	C₂²×C₂₃	1	112	yes	direct product factor
64	C₂×C₂₃	C₂²×#97	1	28	yes	direct product factor
65	C₂×C₂₃	C₂³×C₄²	1	1	yes	characteristic
66	C₂×C₂₃	C₂³×M₁₆	1	2	yes	—
67	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
68	C₂³×C₄	—	2	16	no	—
69	C₂³×C₄	C₂×C₄×C₂₃	1	28	yes	—
70	C₂³×C₄	C₂×C₄×C₂₃	1	7	yes	—
71	C₂³×C₄	C₂×C₄×C₂₃	1	16	yes	—
72	C₂³×C₄	C₂³×C₂₃	1	7	yes	—
73	C₂²×C₈	C₂×C₄×C₂₃	1	224	yes	complemented
74	C₂²×C₈	C₂³×C₂₃	1	56	yes	—
75	C₂×C₄²	C₂×C₄×C₂₃	1	84	yes	—
76	C₂×C₄²	C₂³×C₂₃	1	28	yes	—
77	#97	C₂³×C₂₃	1	64	yes	direct product factor
78	C₂₃	C₂³×#97	1	1	yes	characteristic, direct product factor, Sylow
79	C₂⁴	C₄²×C₂₃	1	8	yes	—
80	C₂⁴	M₁₆×C₂₃	1	16	yes	—
81	C₂⁴	C₂²×C₄×C₂₃	1	7	yes	—
82	C₂²×C₄	—	2	112	no	—
83	C₂²×C₄	C₄²×C₂₃	1	56	yes	—
84	C₂²×C₄	C₂²×C₄×C₂₃	1	56	yes	—
85	C₂²×C₄	C₂²×C₄×C₂₃	1	21	yes	—
86	C₂²×C₄	C₂²×C₄×C₂₃	1	56	yes	—
87	C₂²×C₄	C₂⁴×C₂₃	1	7	yes	—
88	C₂×C₈	C₂²×C₄×C₂₃	1	224	yes	complemented
89	C₂×C₈	C₂⁴×C₂₃	1	16	yes	—
90	C₄²	C₂²×C₄×C₂₃	1	56	yes	—
91	C₄²	C₂⁴×C₂₃	1	8	yes	—
92	C₂³	C₂₃×#97	1	64	yes	direct product factor
93	C₂³	C₂×C₄²×C₂₃	1	28	yes	—
94	C₂³	C₂×M₁₆×C₂₃	1	56	yes	—
95	C₂³	C₂³×C₄×C₂₃	1	7	yes	—
96	C₂×C₄	—	2	112	no	—
97	C₂×C₄	C₂×C₄²×C₂₃	1	84	yes	—
98	C₂×C₄	C₂³×C₄×C₂₃	1	16	yes	—
99	C₂×C₄	C₂³×C₄×C₂₃	1	28	yes	—
100	C₂×C₄	C₂³×C₄×C₂₃	1	7	yes	—
101	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
102	C₈	C₂³×C₄×C₂₃	1	32	yes	complemented
103	C₂²	C₂×C₂₃×#97	1	112	yes	direct product factor
104	C₂²	C₂²×C₄²×C₂₃	1	14	yes	—
105	C₂²	C₂²×M₁₆×C₂₃	1	28	yes	—
106	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
107	C₄	—	2	16	no	—
108	C₄	C₂²×C₄²×C₂₃	1	14	yes	—
109	C₄	C₂⁴×C₄×C₂₃	1	2	yes	—
110	C₂	C₂²×C₂₃×#97	1	28	yes	direct product factor
111	C₂	C₂³×C₄²×C₂₃	1	1	yes	characteristic
112	C₂	C₂³×M₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₂₃	C₂²
commutator	C₂	C₂³×C₄²×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle, prime radical	C₂⁵×C₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂²×C₂₃×#97	C₂	direct (N × H)
6. C₂³×C₈×C₂₃	C₄	semidirect
10. C₂×C₂₃×#97	C₂²	direct (N × H)
17. C₂²×C₈×C₂₃	C₂×C₄	semidirect
21. C₂₃×#97	C₂³	direct (N × H)
31. C₂×C₈×C₂₃	C₂²×C₄	semidirect
35. C₂³×#97	C₂₃	direct (N × H)
36. C₂³×C₂₃	#97	direct (N × H)
46. C₈×C₂₃	C₂³×C₄	semidirect
49. C₂²×#97	C₂×C₂₃	direct (N × H)
50. C₂²×C₂₃	C₂×#97	direct (N × H)
59. C₂³×C₈	C₄×C₂₃	semidirect
63. C₂×#97	C₂²×C₂₃	direct (N × H)
64. C₂×C₂₃	C₂²×#97	direct (N × H)
73. C₂²×C₈	C₂×C₄×C₂₃	semidirect
77. #97	C₂³×C₂₃	direct (N × H)
78. C₂₃	C₂³×#97	direct (N × H)
88. C₂×C₈	C₂²×C₄×C₂₃	semidirect
92. C₂³	C₂₃×#97	direct (N × H)
102. C₈	C₂³×C₄×C₂₃	semidirect
103. C₂²	C₂×C₂₃×#97	direct (N × H)
110. C₂	C₂²×C₂₃×#97	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₈×C₂₃	2
C₂³×C₄²×C₂₃	2
C₂²×C₂₃×#97	2
C₂³×#97	23

Subgroups — C23×C23×#97 #19207677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#97 #19207677